When considering functions of several variables, Korányi and Pukánszky gave an integral representation for holomorphic functions on the unit polydisk with non-negative real part [1]}. Additionally, they gave a criterion to determine when a function defined a priori on a subset of \(n\)  of a particular type can be extended to holomorphic function on the whole of \(n\)  with non-negative real part [1]}. The latter condition of the latter theorem is expressed in terms of a certain function being positive semi-definite. Interestingly, this positive semi-definite function turns out to be a real constant multiple of the one appearing in the following adaptation of Theorem REF .
